
(Replacing the truncation parameter) The truncation parameter \(k_n\)  as defined in (REF ) depends on the unobservable population eigenvalues \(\lambda _j\) 's. The natural substitute is the empirical truncation
\(\hat{k}_n := \max \lbrace  p = 1, \ldots , n : \hat{\lambda }_p + \hat{\delta }_p / 2 \ge c_n \rbrace ,\) 
where \(\hat{\delta }_j\)  is as analogously defined to its population counterpart in (REF ) but with the empirical eigenvalues. Thanks to Assumption REF (ii) and [1]}, we have \(\sup _{j \ge 1} |\hat{\lambda }_j - \lambda _j| \rightarrow 0\)  almost surely. Hence for sufficiently large sample sizes, using the empirical truncation \(\hat{k}_n\)  or population truncation \(k_n\)  are equivalent in probability.

(Replacing the optimization domain) The optimization domain as defined in (REF ) is over the unobservable population eigenvectors \(e_j\) 's. The natural empirically feasible approach is to optimize instead over the empirical eigenvectors \(\hat{e}_j\) 's. By Assumption REF , [1]} ensures \(\mathbb {E}\left[ \sup _{1 \le j \le k_n} \vert \vert \hat{e}_j - e_j^{\prime } \vert \vert ^2 \right] \rightarrow 0\)  as \(n \rightarrow \infty \) , and where we have denoted \(e_j^{\prime } := \mathrm {sign}(\left\langle \hat{e}_j,e_j \right\rangle ) e_j\)  and where \(\mathrm {sign}(t) = 1\)  if \(t > 0\) , \(= 0\)  if \(t = 0\) , and \(= -1\)  if \(t < 0\) . This implies optimizing over \(\hat{\mathcal {J}}_n:= \mathrm {ball}\,{H}\cap \mathrm {span}\lbrace  \hat{e}_1, \ldots , \hat{e}_{k_n} \rbrace \)  and \(\mathrm {ball}\,{H}\cap \mathrm {span}\lbrace  e_1^{\prime }, \ldots , e_{k_n}^{\prime } \rbrace \)  are asymptotically equivalent in probability. By fixing the “orientations” \(\mathrm {sign}(\left\langle \hat{e}_j,e_j \right\rangle )\) 's, we can identify optimizing \(\mathrm {ball}\,{H}\cap \mathrm {span}\lbrace  e_1^{\prime }, \ldots , e_{k_n}^{\prime } \rbrace \)  with optimizing over \(\mathcal {J}_n\) .

(Replacing the asymptotic standard deviation) The asymptotic standard deviation \(t_n\)  as defined in (REF ) depends on the unobservable population eigenvalues \(\lambda _j\) 's and eigenvectors \(e_j\) 's. An empirically feasible version of \(t_n\)  is its natural plug-in estimator,
\(\hat{t}_n(h) =\sqrt{ \sum _{j = 1}^{\hat{k}_n} \hat{\lambda }_j [f_n(\hat{\lambda }_j)]^2 \left\langle h,\hat{e}_j \right\rangle ^2 } + a_n,\quad h \in \hat{\mathcal {J}}_n\) 
By using the arguments in (a) and (b) above, it is not difficult to see that \(t_n\)  and \(\hat{t}_n\)  are asymptotically the same in probability. See also [3]}.

(Consistent estimate of noise error) It is clear the standard deviation of the error term \(\sigma _\varepsilon \)  can be replaced by any consistent estimator \(\hat{\sigma }_\varepsilon \overset{\mathbb {P}}{\rightarrow } \sigma _\varepsilon \) .


